Jon Bannon, Donald Hadwin and Maureen Jeffery

Abstract

By a result of the second author, the Connes embedding conjecture (CEC) is
false if and only if there exists a self-adjoint noncommutative polynomial
p(t1,t2) in the universal
unital
C∗-algebra
A=〈t1,t2:tj=tj∗,0<tj≤1for1≤j≤2〉 and positive, invertible
contractions
x1,x2 in a finite
von Neumann algebra
ℳ
with trace
τ
such that
τ(p(x1,x2))<0 and
Trk(p(A1,A2))≥0 for every positive
integer
k and all positive
definite contractions
A1,A2
in
Mk(ℂ). We prove
that if the real parts of all coefficients but the constant coefficient of a self-adjoint polynomial
p∈A have the same sign, then
such a
p cannot disprove
CEC if the degree of
p
is less than
6,
and that if at least two of these signs differ, the degree of
p is
2, the coefficient
of one of the
ti2
is nonnegative and the real part of the coefficient of
t1t2 is zero then
such a
p
disproves CEC only if either the coefficient of the corresponding linear term
ti is nonnegative or both
of the coefficients of
t1
and
t2
are negative.